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Cauchy Sequences of Real Numbers

One very important classification of sequences are known as Cauchy Sequences which we defined as follos:

Definition: A sequence $(a_n)$ is called a Cauchy Sequence if $\forall \epsilon > 0$ there exists an $N \in \mathbb{N}$ such that $\forall m, n ≥ N$, then $\mid a_n - a_m \mid < \epsilon$. A sequence is not Cauchy if $\exists \epsilon_0 > 0$ such that $\forall N \in \mathbb{N}$ there exists at least one $m$ and one $n$ where $m, n > N$ such that $\mid a_n - a_m \mid ≥ \epsilon_0$.

As you might suspect, if $(a_n)$ and $(b_n)$ are Cauchy sequences, then the sequences $(a_n + b_n)$, $(a_n - b_n)$, $(ka_n)$ and $(a_nb_n)$ are also Cauchy. The proofs of these can be found on the Additional Cauchy Sequence Proofs page.

Lemma 1: Every convergent sequence $(a_n)$ of real numbers is also a Cauchy sequence.

Proof: Let $(a_n)$ be a convergent sequence to the real number $A$. Then $\forall \epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - A \mid < \epsilon$. So, for $\epsilon_1 = \frac{\epsilon}{2} > 0$ there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - A \mid < \epsilon_1 = \frac{\epsilon}{2}$ and so if $m, n ≥ N$ then: